The equations of general Hassett maximal cubic fourfolds

Technical Summary: The Equations of General Hassett Maximal Cubic Fourfolds

Problem Statement
Cubic fourfolds occupy a central position in modern algebraic geometry due to their complex Hodge theory and the unresolved rationality problem. A cubic fourfold $X \subset \mathbb{P}^5$ is termed "special" if the rank of its algebraic cohomology lattice $A(X)$ is at least 2. Hassett established that the locus of special cubic fourfolds forms a union of irreducible divisors $C_d$, indexed by a discriminant $d$ belonging to the Hassett subset $\mathcal{H} = \{n \in \mathbb{N}_{\ge 8} : n \equiv 0, 2 \pmod 6\}$.

The intersection of all such Hassett divisors, denoted $Z = \bigcap_{d \in \mathcal{H}} C_d$, represents cubic fourfolds with maximally rich geometric structures, termed "Hassett maximal." In prior work, the authors (Gal and Nuer) established that the dimension of $Z$ is 16 and that any irreducible component of $Z$ corresponds to a moduli space $C_M$ of $M$-polarizable fourfolds, where $M$ is a positive-definite lattice of minimal rank (rank 4) primitively representing $\mathcal{H}$. However, the explicit geometric relationship between the lattice $M$ and the defining equations of these fourfolds remained unclear. The primary objective of this note is to construct an explicit family of cubic polynomials whose vanishing loci constitute a 16-dimensional irreducible component of $Z$, thereby providing a concrete geometric description of general Hassett maximal cubic fourfolds.

Methodology
The authors employ a combination of geometric construction, lattice theory, and arithmetic methods to achieve their goals:

1. Geometric Construction via Plane Arrangements:
   Leveraging a foundational result by Voisin regarding the intersection of planes in cubic fourfolds, the authors define a specialized family $\mathcal{N}$ of cubic fourfolds containing four planes ($P_1, P_2, P_3, P_4$) with a specific incidence profile:

   • $P_1 \cap P_2$ and $P_1 \cap P_3$ are lines.
   • $P_2 \cap P_3$ is a point.
   • $P_1 \cap P_4 = \emptyset$.

   Using the incidence relations, the authors derive the defining ideals for these planes in terms of linear forms. By analyzing the intersection of these ideals, they construct a specific family of cubic polynomials $F$ that vanish on all four planes. This leads to an explicit algebraic form for $F$ involving parameters $a, b \in \mathbb{C}$.

2. Dimensional Analysis:
   The authors analyze the parameter space of such plane arrangements modulo the action of $PGL(6, \mathbb{C})$. By calculating the dimension of the incidence correspondence between plane arrangements and cubic fourfolds, they demonstrate that the resulting family $\mathcal{N}$ has dimension 16, matching the known dimension of the Hassett maximal locus $Z$.

3. Arithmetic Verification via ADC Forms:
   To prove that this constructed family $\mathcal{N}$ indeed covers the entire Hassett maximal locus (i.e., that the primitive image of the associated lattice spans the entire Hassett subset $\mathcal{H}$), the authors analyze the associated rank-4 lattice $L(M)$. They reduce the problem to showing that a specific positive-definite quadratic form $F$ represents every integer in $\mathcal{H}$ primitively.

   This requires proving that the form $F$ (and related ternary forms) possesses the ADC property (Any integer representable by a rational vector is also representable by an integer vector). The authors:

   • Establish that the ternary forms $Q_3(x, y, z) = x^2 + y^2 + 3z^2$ and $G(x, y, z) = x^2 + 3y^2 + 3z^2$ are ADC forms.
   • Utilize a generalized geometric descent argument (extending the Davenport-Cassels method via torus partitioning) to confirm the ADC property for non-Euclidean forms.
   • Use the Hasse-Minkowski theorem and local-global principles to demonstrate that the image of the form covers all necessary residues modulo specific integers, ensuring that every element in $\mathcal{H}$ is primitively represented.

Key Contributions and Results

• Explicit Equations: The paper provides the first explicit construction of a family of cubic fourfolds that are Hassett maximal. The general member of this family is isomorphic to the vanishing locus of a homogeneous cubic polynomial $F$ belonging to the intersection of four specific ideals:
  $$F \in (x, y, z) \cap (x, y, u) \cap (x, z, v) \cap (v - by, u - az, w)$$
  where $x, y, z, u, v, w$ are linear forms and $a, b \in \mathbb{C}$ are constants.
• Dimensional Confirmation: The authors prove that the locus $\mathcal{N}$ defined by these equations has dimension 16, confirming it constitutes an irreducible component of the Hassett maximal locus $Z$.
• Lattice Maximality: By proving the ADC property for the associated quadratic forms, the authors demonstrate that the primitive image of the lattice $L(M)$ associated with these fourfolds is exactly the Hassett subset $\mathcal{H}$. This confirms that the constructed family is indeed "Hassett maximal," meaning these fourfolds lie in the intersection of all Hassett divisors $C_d$.
• Arithmetic Tool: The paper contributes a refined geometric descent argument (Appendix A) to confirm the ADC property for specific non-Euclidean ternary forms, a technical result used to bridge the gap between the lattice construction and the arithmetic properties of the Hassett subset.

Significance
The paper claims to resolve the "opaque" relationship between the abstract lattice $M$ and the concrete geometry of Hassett maximal cubic fourfolds. By providing explicit polynomial equations, the authors move the study of these maximal objects from a purely theoretical existence result to a concrete, computable framework. The work confirms that the 16-dimensional family of cubic fourfolds defined by the intersection of specific plane ideals is not only of the correct dimension but also arithmetically complete with respect to the Hassett subset, thereby fully characterizing a general component of the Hassett maximal locus.